Pinning synchronization of unilateral coupling neuron network with stochastic noise
Introduction
Synchronization is ubiquitous in nature and plays an important role in many fields, such as biology, ecology [1], [2]. Various synchronizations are observed in biological experiments and numerical simulations, for example, mutual synchronization [3], entrainment and chaotic synchronization [4]. With the development of nonlinear dynamics, the classical concept of synchronization has been extended from the phase locking of periodic oscillators to that of chaotic oscillators. Many kinds of synchronizations are described, e.g., complete synchronization, phase synchronization, generalized synchronization, etc. [5], [6], [7]. So far, many approaches have been proposed for synchronization of chaotic systems, for instance, linear and nonlinear feedback synchronization method [8], [9], adaptive synchronization method [10], time-delay feedback approach [11], backstepping design method [12], sliding mode control method [13], impulsive synchronization method [14], etc. Most of the existing methods can synchronize two identical or different low-dimensional chaotic systems.
In the past decade, complex network attracted more and more attention of researchers. It is partially due to the fact that any large-scale and complicated system in real world can be modeled by a complex network, in which nodes can be seen as the elements of the system and edges can be considered as interactions between nodes. For example, the WWW, the Internet, neural network, social network and cited network are all complex networks. To master the complicated nature of complex network, more and more people began to investigate dynamics of complex network, such as robustness [15], pinning synchronization [16], [17].
As a complex network, neural network exhibits collective dynamics, which give us insight into the mechanism of information processing and transfer. This fact makes it interesting to study the coupled neurons in recent years. The phenomenological neuron model proposed by Hindmarsh and Rose [18] may be seen either as a generalization of the Fitzhugh equations [19] or as a simplification of the physiologically realistic model proposed by Hodgkin and Huxley [20]. Hindmarsh–Rose (HR) model is a single compartment model and is able to reproduce all the dynamical behaviors of neural network. To understand it, many people explored the dynamics of neural networks [21], [22]. Recently, kinds of synchronization in the model of HR neurons have been extensively investigated, such as robust synchronization [23], spike phase synchronization [24], adaptive synchronization [25].
Inspired by above work, pinning synchronization of unilateral coupling complex neural network with stochastic noise is investigated in this paper. Other parts of the paper are arranged as follows. HR neuron model is presented in Section 2. In Section 3, schemes are given to realize the pinning synchronization between the given reference signal and time-delay HR neuron system with stochastic noise. To verify the theoretical results, numerical simulations are given in Section 4. Some conclusions are drawn in Section 5.
Section snippets
Unilateral coupling Hindmarsh–Rose (HR) neural network with time delay and stochastic noise
In this paper, HR neuronal model with time-delay is considered as follows:where τ > 0 is the time delay. When τ = 0, model (1) is HR neuronal model proposed by Hindmarsh and Rose as a mathematical representation of the firing behavior of neurons [18]. It was originally introduced to give a bursting type with long interspike intervals of real neurons. In real neuron system, time-delay always exists when signals are communicated among
Control scheme
In this section, with the aid of appropriate controller, the pinning synchronization of complex HR neuronal network is investigated. For any given reference signal s(t), pinning synchronization between system (2) and s(t) can be obtained. To this end, the controlled unilateral coupling neural network with time delay and stochastic noise is expressed as follows
Simulation
In this section, simulations are given to illustrate the effectiveness of the proposed method. In simulations, coupling intensity Hi (i = 2, …, N) is taken as the same value. The original values are taken as constants. Each node is a HR system. The reference signal s(t) is taken as the behavior of an isolated HR neural node. The parameters are chosen as a = 3.0, b = 1.0, c = 1.0, d = 5.0, r = 0.006, S = 4.0, k = 1.6, Iext = 3.1, and the time delay is chosen as τ = 1, with which the HR system is chaotic bursting. To
Conclusion
In this paper, based on Lyapunov stability theory, scheme to achieve the pinning synchronization of unilateral coupling neural network with time delay and stochastic noise is proposed. We not only test the validity of the proposed scheme from theoretical analysis, but also verify the effectiveness of the scheme from numerical simulations. Furthermore, from the simulations, some results can be known as follows:
- (1)
When coupling intensity and noise intensity are constants, the time of convergence is
Acknowledgements
This work is supported by National Natural Science Foundation of China (Grant Nos. 11102180, 61379064 and 61273106), the Natural Science Foundation of the Jiangsu Province of China (Grant No. BK2012672), and the Qing Lan Project.
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